#102670

order	512
nilpotency class	3
isoclinism	isoclinic class of #102649
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

499 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,15)(13,17)(16,21)(18,22)(19,24)(20,23)(26,31)(27,34)(28,29)(33,38)(35,39)(36,37);(5,8)(9,11,13,19)(10,14,17,24)(12,20,21,18)(15,23,16,22)(25,26,30,33)(27,35,37,39)(28,38,29,31)(32,34,40,36);(1,3)(2,5)(4,7)(6,8)(10,15)(12,21)(14,22)(16,17)(18,20)(23,24)(25,27,30,37)(26,32,33,40)(28,34,29,36)(31,39,38,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=egi,f^b=fg,d^c=dh,e^c=ei,f^c=fg

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9866);